I have a Spring application in a Tomcat using the AJP connector with Apache httpd so that I can access directly the application through: https://example.com instead of https://example.com:9700.
The Spring application uses Spnego for kerberos communications.
Does the outgoing call pass through the HTTP Apache Server or will Tomcat handle it itself?
Outgoing connections have nothing to do with incoming ones, so the answer is no.
Unless you or your network happens to (configure or) force you to go through a proxy, the outgoing connection is directly between Tomcat and your target.
Why would there be a connection between "incoming through AJP" and "outgoing through http" (or through any protocol) anyway? AJP is just good for forwarding http, not as a generic VPN-like proxy.